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Journal of Mathematical Physics

Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure: Part I

Renzo Cirelli, Alessandro Manià, Livio Pizzocchero B

Published Dec 1, 1990• 99 citations
Publisher

Abstract

Schrödinger quantum mechanics is formulated as an infinite-dimensional Hamiltonian system whose phase space carries an additional structure (uncertainty structure) to account for the probabilistic character of the theory. The algebra of observables is described as an algebra of smooth functions on the quantal phase space, with a product naturally induced by the geometrical structures living on that manifold. The possibility of generalizing Schrödinger mechanics along these lines is discussed.

DOI
10.1063/1.528941

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Schrödinger quantum mechanics is formulated as an infinite-dimensional Hamiltonian system whose phase space carries an additional structure (uncertainty structure) to account for the probabilistic character of the theory. The algebra of observables is described as an algebra of smooth functions on the quantal phase space, with a product naturally induced by the geometrical structures living on that manifold. The possibility of generalizing Schrödinger mechanics along these lines is discussed.
We develop a geometrical approach to Schrödinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kähler structure possessed by the set of the pure states of a quantum system. The Kähler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kähler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrödinger quantum mechanics.
Constructing a classical mechanical system associated with a given quantum-mechanical one entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum-mechanical system naturally has the structure of an infinite-dimensional symplectic manifold ('quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straightforwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space . We report on how the procedure also allows extraction of nonlinear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin- j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space mathbbRmathrm2N\\mathbb{R}^{\\mathrm{2N}}mathbbRmathrm2N. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Bohr quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality. There is also an alternative and a priori distinct notion of effective Hamiltonian used in the context of a WKB approximation. These two notions are compared for the example of loop quantum cosmology.
This work develops a unified structural explanation of Heisenberg’s uncertainty principle based onthe infinite-dimensional nature of physical reality and its mathematical representation. Quantumstates are intrinsically wave-like and must be described by functions rather than finite-dimensionalvectors, forcing the use of an infinite-dimensional Hilbert space. Within this framework, fourindependent but mutually reinforcing structures emerge: analytic non-commutativity ofmultiplication and differentiation, Fourier duality and the mutual broadening of a function and itstransform, the symplectic geometry of phase space and its minimal area constraint, and thealgebraic rigidity encoded in the Weyl relations and the Stone–von Neumann theorem. Theselayers form a logical chain from finite-dimensional impossibility to infinite-dimensional necessity,revealing the uncertainty principle as a structural feature of the physical world. The paper furtherdiscusses extensions to the generalized uncertainty principle and the infinite-dimensionalstructure of quantum field theory.
In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum observables to real functions on P(H), classical-like observables, and quantum states to probability densities on P(H), Liouville densities, in order to obtain a complete and meaningful Hamiltonian formulation of a finite-dimensional quantum theory.